The Leggett-Garg inequality, which is an analogue of Bell’s inequality
that involves correlations of measurements on a system at different
times, is one of the hallmark tests of quantum mechanics against
classical predictions. According to Formaggio et
al. the phenomenon of
neutrino oscillations should adhere to predictions of quantum-mechanics
and provide an observable violation of the Leggett-Garg inequality. In
this paper Formaggio et al.
demonstrate how oscillation phenomena can be used to test for violations
of the classical bound by carrying out measurements on neutrinos at
distinct energies, instead of a single neutrino at distinct times. It is
shown by a study of the data provided by the MINOS experiment that there
is a greater than 6σ violation over a distance of 735 km, which
represents the greatest distance over which either the Leggett-Garg
inequality or Bell’s inequality has been tested.

The principle of superposition is possibly one of the counterintuitive
aspects of quantum mechanics, which stipulates that an entity can exist
in multiple different states simultaneously. It has been indicated by
Bell and others how distinguishing between classical systems and those
demonstrating quantum superposition could be shown experimentally (Bell,
1964; Kochen & Specker, 1967). Bell’s inequality concerns correlations
among measurements on systems that are separated spatially. Analogous
tests were developed by Leggett and Garg that concerns correlations
among measurements that have been performed on a system at different
times, and they extended the tests to apply to macroscopic entities
(Leggett & Garg, 1985). The Leggett-Garg inequality (LGI), referred to
sometimes as the “time-analogue” of Bell’s inequality, allows for a
complementary test of quantum mechanics while potentially avoiding some
of the difficulties that are involved in performing a test that is truly
free of loop holes of Bell’s inequality (Hensen et al., 2015; Shalm et
al., 2015; Giustina et al., 2015; Gallicchio, Friedman & Kaiser, 2014).
See (Emary, Lambert & Nori, 2014) for a recent review.